What's wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces

Abstract

A celebrated result of Mirzakhani states that, if $(S,m)$ is a finite area \emph{orientable} hyperbolic surface, then the number of simple closed geodesics of length less than $L$ on $(S,m)$ is asymptotically equivalent to a positive constant times $L^{\dim\mathcal{ML}(S)}$, where $\mathcal{ML}(S)$ denotes the space of measured laminations on $S$. We observed on some explicit examples that this result does not hold for \emph{nonorientable} hyperbolic surfaces. The aim of this article is to explain this surprising phenomenon. Let $(S,m)$ be a finite area \emph{nonorientable} hyperbolic surface. We show that the set of measured laminations with a closed one--sided leaf has a peculiar structure. As a consequence, the action of the mapping class group on the projective space of measured laminations is not minimal. We determine a partial classification of its orbit closures, and we deduce that the number of simple closed geodesics of length less than $L$ on $(S,m)$ is negligible compared to $L^{\dim\mathcal{ML}(S)}$. We extend this result to general multicurves. Then we focus on the geometry of the moduli space. We prove that its Teichm\"uller volume is infinite, and that the Teichm\"uller flow is not ergodic. We also consider a volume form introduced by Norbury. We show that it is the right generalization of the Weil--Petersson volume form. The volume of the moduli space with respect to this volume form is again infinite (as shown by Norbury), but the subset of hyperbolic surfaces whose one--sided geodesics have length at least $\varepsilon>0$ has finite volume. These results suggest that the moduli space of a nonorientable surface looks like an infinite volume geometrically finite orbifold. We discuss this analogy and formulate some conjectures.